Thickness in Go describes a strong formation of stones. A heavy group is one that's over-concentrated. How do I know when my stones are thick, without going too far and becoming heavy? It seems like this can happen in just a very few moves. Are there any good tutorials that illustrate positions where a player turns a thick group into a heavy one?
You know your stones are thick when they form a safe group that secures some territory and/or build some influence, threatens to attack... or just give you some kind of advantage.
There is no official definition for "heavy stones", but I think the term was used instead of the term bad shape for a group of 5 stones or more.
In any case, if you have a group of stones all connected together that doesn't do much to secure some territory or reduce the opponent's, then you probably have a bad, heavy, over concentrated shape.
Thickness is indeed about the same as strength and comes with "eye potential" and "connectivity". We only speak of thickness in the context of outward influence, while strong enclosed groups or groups with limited potential for expansion are rather called "solid".
Most of the answers get this part right. But "heavy" is not at all the same as "overconcentrated".
Heavy stones are never thick: they may be large in number, but they lack eye potential, sometimes even making dumpling shapes. They are weak and vulnerable. More importantly, they are not easy to sacrifice, so they place a burden on the player.
Overconcentrated stones on the other hand can be and often are thick: if a player has created 3 or more eyes for his group, such investment was exaggerated, hence the stones are called overconcentrated.
Now if you start to fill all your eyes, turning it into a dumpling shape, a thick, overconcentrated group will become heavy indeed. But this would really be a novice mistake.
Next to that, it is imporant to understand that "being thick, heavy or overconcentrated" can be an aspect of a local group, but also of the entire distribution of stones. "Black is thick" means there are no easy ways to attack him. "White is heavy" means White has lots of burden. "Black is overconcentrated" means he has invested too much in a few areas.
Thickness and heavy stones are "similar," insofar as they involve a large number of stones. But there the similarity ends, because thick and heavy groups function differently.
Thick stones are those that you want to drive your opponent toward, because they are secure, meaning that your opponent's relatively few stones will suffer in the presence of your many.
Heavy stones are sometimes referred to as "dangos" or "clumps," awkward, insecure groups of stones that are easily chased by your opponent's stones, even if fewer. They are stones you'd rather have your opponent stay away from.
Occasionally, if a group of thick stones have suddenly lost their "eye space" or foundation, they can become heavy. On the other hand, a heavy group with new-found eyes can become thick.
The terms thickness and influence are precisely defined and very easy to differentiate once you got the idea.
Influence simply refers to a group facing the center of the board.
If a group create influence and at the same time is already alive or can make eyes very easily is called thick. Note that a living group that is facing the edge of the board is not thick.
Distinguishing between thickness and heaviness or overconcentration is more difficult.
Heaviness refers to a group of many stones yet few eyespace. Typically, it is created by trying to save too many stones, or by playing bad shape.
Overconcentration is a different phenomenon and refers to cramped positions, typically hand in hand with a violation of the wall+1 rule. Overconcentrated positions tend to be extremely thick, far past the peak of efficiency.
I'm no expert (10k on a good day) but if you follow the general rule of playing away from thickness you should be off to a good start. If you fail to play far enough away you could easily get over connected which could lead to a heavy group.